↑

Properties

Label 2-735-105.53-c1-0-13
Degree $2$
Conductor $735$
Sign $-0.912 + 0.408i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.340 + 1.26i)2-s + (0.664 + 1.59i)3-s + (0.236 + 0.136i)4-s + (−1.25 + 1.85i)5-s + (−2.25 + 0.299i)6-s + (−2.11 + 2.11i)8-s + (−2.11 + 2.12i)9-s + (−1.92 − 2.21i)10-s + (3.38 + 1.95i)11-s + (−0.0611 + 0.468i)12-s + (1.56 + 1.56i)13-s + (−3.79 − 0.767i)15-s + (−1.69 − 2.92i)16-s + (2.58 − 0.693i)17-s + (−1.97 − 3.41i)18-s + (1.61 − 0.930i)19-s + ⋯
L(s)  = 1  + (−0.240 + 0.897i)2-s + (0.383 + 0.923i)3-s + (0.118 + 0.0681i)4-s + (−0.559 + 0.829i)5-s + (−0.921 + 0.122i)6-s + (−0.746 + 0.746i)8-s + (−0.705 + 0.708i)9-s + (−0.609 − 0.701i)10-s + (1.01 + 0.588i)11-s + (−0.0176 + 0.135i)12-s + (0.434 + 0.434i)13-s + (−0.980 − 0.198i)15-s + (−0.422 − 0.731i)16-s + (0.627 − 0.168i)17-s + (−0.466 − 0.803i)18-s + (0.369 − 0.213i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.912 + 0.408i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.912 + 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304354 - 1.42506i\)
\(L(\frac12)\) \(\approx\) \(0.304354 - 1.42506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.664 - 1.59i)T \)
5 \( 1 + (1.25 - 1.85i)T \)
7 \( 1 \)
good2 \( 1 + (0.340 - 1.26i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-3.38 - 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.56 - 1.56i)T + 13iT^{2} \)
17 \( 1 + (-2.58 + 0.693i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.61 + 0.930i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.38 + 0.638i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.513T + 29T^{2} \)
31 \( 1 + (-4.29 + 7.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.60 + 1.77i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.308iT - 41T^{2} \)
43 \( 1 + (-7.60 - 7.60i)T + 43iT^{2} \)
47 \( 1 + (-1.36 + 5.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.498 - 1.85i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.259 - 0.448i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.55 - 4.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.34 - 8.74i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 15.3iT - 71T^{2} \)
73 \( 1 + (2.79 - 0.749i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.37 + 2.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.16 - 9.16i)T - 83iT^{2} \)
89 \( 1 + (-5.67 - 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.81 + 6.81i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.80914251797000094277871521374, −9.820802891590676019554977003463, −9.083292896090453202572305814017, −8.164258441045764571484328912926, −7.46221469671411056079022389268, −6.60393253183691896514207407990, −5.74881045320399640901853588725, −4.39455497004083114337615016010, −3.53228819228247012287378144255, −2.42781837981987311047892416527, 0.818798285668638722088885470978, 1.61472303322028044146280723606, 3.13463967184540814267907784763, 3.83489051528170233039885337956, 5.53972187275420479046540253043, 6.41785692604114288364026626166, 7.39712926963491380826387082529, 8.437003336431184639208724159297, 8.919973982180571994096992546686, 9.886920783805841142583253492399

Graph of the $Z$-function along the critical line